Result 3: The Damper-Only Model

The Model ------------------------------------------------

<Ordering Policy>

The amount of order at time t is a linear function of the change in inventory level. That is,

Order(t) = c [Inventory(t-1) - Inventory(t)] ,if Inventory(t-1) - Inventory(t) > 0
Order(t) = 0 ,Otherwise

where c = sensitivity to the change in inventory. (Note: Backlog = - Inventory)

<Consumer Demand>

4 until week 4. Random in [0, 16] after week 4

Result --------------------------------------------------------
<c = 0.1 (small)>

The inventory levels of retailer and wholesaler (negatively) explode and the system goes unstable. Why? Because this is a PURE dash-pot model. Imagine dropping a stone in the ocean - yes, there is resistance that is proportional to the stone's speed, but it keeps sinking anyway. That's what's happening here.

Likewise, if c is too large...

The inventory levels upstream keep accumulating. Again, the system is unstable.

However, c is not too small or too large...

Notice how beautifully the four curves move together - even between retailer and factory! And there seems to be an attractor.

Then, the next question is, what's the range of c that makes this possible?

The figure below shows that range:

I'm intrigued by this result.

- There is indeed a small range of c that minimizes cost. [1 - 1.6]
- Although it's hard to see in this figure, the minimum cost is about 108 and it's almost constant throughout that range.
- There is phase transition exactly at c = 1.
- There is also a unique c that offers a local maximum around c = 0.65.

You might think that because this range of c is so small and because of the phase transition, the dampening policy is too dangerous. I tend to agree, but there is still a few advantages:

1. If the critical value c = 1 is UNIVERSAL, all you have to do is set your c a bit higher than 1, since there IS at least some range that minimizes the cost.
2. There is only one parameter to adjust. In other words, you don't have to set a target inventory.

Next Step & Discussion ----------------------------------------------------------

The reason too large c or too small c makes the system unstable is that this model is a "pure" damper model. There is no anchoring mechanism. So, if we combine this damper model and the earlier "spring" model, we might be able to stabilize the system more easily, perhaps at a desired inventory level.

This reminds me of Hedley's earlier remark:

> One thing that struck me is the resemblance to something that
>I would have thought to be entirely unrelated - in electronic circuitry,
> the response of the internals of a frequency division system to a noisy input.

I totally agree. In fact, although I had "mechanical" systems in mind, we know there is certain analogy between electrical and mechanical systems. So, in electrical terms, the damper model is like an electrical circuit with resistors and my previous model is that of capacitors or inductors. That's why I think the "combined" model I mentioned above will work. Well, Hedley, you are the specialist, so perhaps you could help us here.

But if this analogy is correct, we might possibly be able to assess the stability of the system analytically too.

Conclusion ----------------------------------------------
The damper model works. But it needs some improvement.